23 January 2007

Flickr Geotags

Flickr is a website that offers free hosting of digital photos. There are some fairly important restrictions on the size of the files that one is allowed to post to flickr without a pro account; for example, nothing over 500 pixels in any direction. Still, Flickr is an amazing website and it offers an amazing amount of value, especially for bloggers (the fact is that, for blogs, there's very seldom any need for bigger photos).

One of my habits is to find photos of things my wife is interested in and email them to her. Today it was photos of Dakar, Senegal. But I noticed that Flickr was showing off a new feature, called "geotags." Basically, you search for a location and Geotag displays the map, with photos of the relevant geographical tag.

Incidentally, to get to the screen displayed, just click the image...

You can select one of the magenta dots to show the photo[s] with that location tag..

And you can view the area as a satellite photo.

That's for those of you who perhaps have become blasé about Google Earth.

20 January 2007

The Dynamics of Industrial Choice (3)

In Part 2 we had a brush with the basic classical economic growth model. The model in question has come in for a significant amount of criticism, not least because it assumes constant full employment (in the sense that unemployment is ALWAYS a choice people make), no significant effect of monopolies, and unlimited ability to defer consumption.

An issue I have with the model is that, ironically, it suppresses the truth it reveals. Technology it treats as absolutely nothing more than the disaggregated residual of economic growth, after subtracting capital and labor growth. The model mentions firms, but they have no objective function to maximize; they aren't constrained by prior technology; they aren't tied to formats. They simply provide a ratio between inputs and output; but both are homogeneous. In plain English, they assume the economy behaves like a gas, not something lumpy. Usually, when you use a mathematical model, it's simplified because there are obvious limits to the complexity of the math one may do. The model may describe optimal conditions for event x, for proving x is impossible even under the most favorable of conditions. Conversely, one might prove the inevitability of x by demonstrating that, even under the most restrictive of conditions (so restrictive they're practically impossible), x will still happen anyway. A third purpose is to show hypothetical conditions under which x could conceivably happen, even though the frequency of those conditions is subject to further inquiry. The RCK model does none of these things.

However, the RCK model can be modified to depict different things entirely. It is not very good at modeling the aggregate economy.* It is somewhat better as a conceptual tool in explaining the forces acting on actors like firms (not households). Households are too varied in character; they are too numerous; and their dynamics of maximization are incompatible with the assumptions of the RCK model. Firms can be grouped into plausible categories based on stranded costs and capital structure; in contrast, households may or may not be constrained by subsistence constraints, multiple members, perverse incentives, and unknown optimization strategies. On the other hand, firms have clear optimization goals.

Additionally, the optimality analysis for which Frank Ramsey had originally suggested his model, is a more reasonable application of the RCK model anyway. In this case, the object is to evaluate the optimality of decisions, not to make predictions or deductions of the "actually existing" economy. Such optimality information about households is, to reiterate, useless; about firms, it can be used to evaluate policy of firm administration. Corporations, with legal powers of limited liability and access to "capital markets," are, in some senses, surrogates of the state. Banks, for example, are a category of firm who are empowered to create money. Their governance is therefore a valid target of this sort of analysis.

A second modification I would recommend pertains to the objective functions that our economic agents seek to maximize. In the original RCK, if the economic actor is off the sadddle point, then it will increase savings to precisely that level required for path-convergence. Yet the functions of capital accumulation out of personal accumulation are pushing in the opposite direction; the situation can be likened to a pedestrian running frantically up a down escalator. That the escalator always points in the direction away from the equilibrium growth position, is an awkward but inevitable fact of the household savings-consumption equation; the optimization function requires that the household will [on average] react by running up the escalator faster than the escalator is moving. From experience, we know this is not true for firms, whose existence takes a clear trajectory from expansion to stagnation to financial collapse.

Corporations manufacturing a particular item are motivated to maximize earnings out of revenues. But rather than following a steady-state optimization function, they have a trajectory, or path through time. Rather than assuming the corporate management optimizes its capital structure and technology choices for a steady state growth, which is unrealistic, we would instead plot the firm as responding to conditions at any moment based on its optimization function, the capital structure, and stranded costs. Firms achieve optimization by selecting between quality and quantity (i.e., between improving processes and enlarging scale). Improved processes result in at least two positive feedbacks: higher revenues (which may persist, longer than the costs of transition to the improved process did) AND lower stranded costs (since the process is revived repeatedly, so stranded costs are reduced in the process design).

I would expect a mathematical exposition of this would reveal that, where market share is basically fixed, quality and low stranded costs would become the preferred choice; and capital structure would tend towards debt, rather than equity.___________________________________________________SOURCES & ADDITIONAL READING: Douglas J. Puffert, "Path Dependence, Network Form, and Technological Change" (PDF); Kenneth J. Arrow, "Path Dependence & Competitive Equilibrium" (PDF); David F. Weiman, "Building ‘Universal Service’ in the Early Bell System: The Reciprocal Development of Regional Urban Systems and Long Distance Telephone Networks" (PDF).___________________________________________________* That the RCK model does not describe observed reality is demonstrated by empirical comparisons of predictions of inflation, interest rates, prices of specific commodities, and so forth. A survey of the historical evidence is found in Steven M. Sheffrin's Rational Expectations. Moreover, the RCK model predicts fairly rapid rates of convergence for economies with disparate levels of productivity—provided trade barriers are low. Convergence of productivity and capital stocks among the economies of the world have not remotely matched expectations. No one has ever tweaked the RCK parameters or equations to provide reasonably accurate predictions of fluctuations of savings or capital (this has been acknowledged by the PDF files explaining the RCK model, linked in part 2.) The same is true for David Romer's Advanced Macroeconomics. Hence, the RCK does not make reasonably accurate predictions about the performance of different economies in the world.

19 January 2007

Some Remarks on the iPhone (2)

Wikipedia: A pseudo-event is an event or activity that exists for the sole purpose of garnering media publicity and serves little to no other function in real life. Since nothing meaningful actually occurs at the event itself, pseudo-events are considered “real” only after they are viewed through news, advertisements, television, or other types of media.

Promotional Photo of the iPhone

The release of the iPhone was the textbook case of the pseudo-event. The big news never was about the technology; the iPhone's features are actually fairly subtle refinements already found (for the most part) on the Blackberry Pearl and on older PDA's such as the ill-fated Apple Newton. Of course, a huge proportion of technical innovation is the best application of many subtle improvements, but this is fairly straightforward bundling of by-now conventional combinations. Apple's big money maker was the iPod; but it was obvious that Apple was missing out on the intimate computing cross-marketing opportunities as long as it was out of the PDA/cell phone business.

No, the real news was supposed to have been the ability of a new firm to enter, and shake up, the PDA/cell phone sector itself. In reality, Apple has become more of a marketing firm than an industrial one; it had to bundle technologies that are now commonplace, and manufactured by subcontractors just like any other gadget. Between 1995 and 1997, Apple Computer executive Gil Amelio had pursued a strategy of licensing the Mac OS to 3rd party manufacturers. This permission to make Mac clones was too short-lived to get very far; in '97 Steve Jobs returned to Apple and effectively terminated licensing. It seems to me that after Jobs rejected the idea of Apple adding value through technology development, there was little choice but for the firm to focus on being cool. Advertisements for the iMac (1998) and other products abandoned appeals to reason. Instead, we were urged to "think different," and posters used images of Albert Einstein, Jane Goodall, or John Lennon with Yoko Ono.

The obvious absurdity of using Amelia Earhart's image to promote the iMac was exacerbated by the fact that the iMac and subsequent products were actually substantially less useful to users than their immediate predecessors (e.g., the Performa). While the Performa was drab in appearance and very pedestrian in its choice of available technologies, the iMac had no floppy drive or expansion slots; it was supposed to relieve customers of the burden of choice. While the applied art was very dramatic, and won awards for its distinctiveness, it was actually fairly difficult to use. The anti-establishment conceit of the Macintosh had become radical chic at its most crass. Apple would face the very reluctant censure of worker health and safety watchdogs, or of environmentalists, as it lagged behind Compaq and Dell in recycling or toxic materials handling.

The pseudo-event of January 2007 was the self-congratulation of managerial elites in the USA: the market for high-end consumer devices really was susceptible to shakeup. Competition was still alive and well. The customer was sovereign. The risk of monopoly was a myth, and those who begged to differ were just losers, or worse. But Apple was not turning the market for anything on its ear. It was simply perpetuating the product cycle for an appliance-cum-fashion accessory. Beyond that, the release of the iPhone did not prove anything about consumer choices at all. The real struggle—between monopoly PCS's and their contending formats—is off the table.And Cingular is the PCS in question. A partner with AT&T*,

___________________________________________________

*AT&T is actually not the former AT&T broken by the US Supreme Court in 1984; it is actually the former SBC Corp., which acquired the 128-year old dinosaur in 2005, then began using its name about three months later. See Oligopoly Watch (1, 2). Earlier, in 2001, AT&T Wireless (a different company) was created as a joint venture of SBC and Bell South, plus 11 other regional PCS's. One of these core companies was a former mobile phone division of the original Ma Bell telecom monopoly, and for this reason SBC & Bell South decided to name their new wireless venture—with TDMA technology—after the still-famous dinosaur. Later, AT&T Wireless became Cingular Wireless, and merged with the newly re-minted AT&T (formerly SBC).___________________________________________________SOURCES: Digital Trends: Apple iPhone Margin Crawls Toward 50 Pct (18 January 2007); Uncommon Knowledge: The iPhone: the good, the bad.. OK, it's definitely not ugly (14 Jan 2007); BoingBoing: the Roach Motel Business Model (14 January 2007) ; c/Net: Review of the Blackberry Pearl (September 2006);

Basically, Cisco registered "iPhone" as a trademark before Apple did. Cisco has sued Apple over the latter's use and alleged that it had been in negotiations over the use of the name when Jobs announced the new iPhone. Cisco's beef is that Jobs released the product before negotiations were complete, and that Apple attempted to acquire the name in bad faith through a front company.

Cisco obtained the iPhone trademark in 2000 when it acquired Infogear, a small Redwood City, Calif., start-up that developed consumer devices that allowed people to easily access the Internet without a PC. Infogear had actually registered the iPhone trademark in March 1996. Cisco's home networking division, Linksys, has been using the iPhone trademark on a new family of voice over IP phones since early last year, Cisco said. And last month, Linksys expanded the iPhone family with additional products.(CNET "Cisco sues Apple" via BoingBoing)

At the heart of the dispute is the putative ability of Cisco to register what is perilously close to being a common noun. But Apple's position is significantly worse: it argues that it has established an "iFamily" of products, such as the iMac, iPod, and iTunes. Therefore, it has an exclusive right to use the iPhone name, regardless of Cisco's prior use.

Cisco alleges that Apple has set up two shell companies to finagle the trademark away (legal brief). I presume this was based on the idea that Cisco might accept a lower price for the name from a smaller company, or some strategy of flooding the courts with prior use claims

One part of the agreement under negotiation between the firms prior to the product release was some form of interoperability between the two products. This way, the fact that two different telephony products featured the iPhone name would not be misleading. Since the release, negotiations with Cisco have been broken off.

The new name and device represent Apple's strategic shift away from its origins as a personal computing company that has at points struggled both to survive and to set the computing world's agenda. The shift was enabled by the five-year-old iPod line of digital media products, which have produced enormous sales and profit growth, propelled Apple into the forefront of the digital media age, and now leave it poised to set the wireless phone industry on its ear. "This is a day I've been looking forward to for two and a half years," Apple CEO Steve Jobs told the capacity crowd at the MacWorld Expo trade show in San Francisco. "Every once in a while a new product comes around that changes everything."

Urm, right. Look, I really do not want to be a curmudgeon, but first—Apple is essentially turning into a VAR, not "the forefront of the media age. " The iPhone is essentially a Blackberry-iPod combination, with some sensors attached:

"We've also got some stuff you can't see—3 advanced sensors. It's got a proximity sensor, bring the iPhone to your ear and your display shuts off and touchscreen shuts down. Ambient light sensor—adjusts brightness, saves power. Third thing is an accelerometer, it can tell whether you're in landscape and portrait."

It's true that combining an iPod with a Blackberry is not as easy as it might seem. Cramming even more features into a tiny object makes the interface crucial; as, for example, since there are not actual buttons, but a touch-sensitive LCD, it's important for the "keypad" to not interface with your ear. And since the phone and iPod combination is new, the music must not interfere with the ring tone (wouldn't it be awful if you were rocking out to the song that also supplies your ring tone?).

It's easy to make fun of the hype surrounding the iPhone, especially (?) from seasoned business journalists like the one who penned the hyperventillating BW article quoted above. For one thing, Hesseldahl is so obsessed with gizmos and justifying monopoly, he swoons over the Apple name and its kewl interface. Seriously, there are trade shows each year with spiffy new technology; saying that an elegant interface will "change everything" is just silly. Show me a transformer that remains efficient over very broad load ranges, and I'll be impressed. This is just a glowing yo-yo.

In truth, the handset makers aren't entirely to blame for the poor customer experience so typical of the mobile-phone industry. Not only do the carriers control the buying experience (something Nokia is trying to change, see BusinessWeek.com, 6/28/06, "Nokia's Ritzy Flagship in Chicago"), the services, the network that determines the speed and kind of services that can be delivered, and the customer service, they also flex their muscle when it comes to the handset. By the time the handset makers and the carriers have fought out the fine points of a design that will work with the network, and the services that will drive revenue, the user's needs have long been forgotten.

Now Apple must join with one of those very carriers, and its choice, Cingular, has already proven somewhat controversial with customers unimpressed with its existing service. And while Apple undoubtedly retains the upper hand, the partnership requires Apple Chief Executive Officer Steve Jobs to loosen his famously tight grip. Apple won't have the end-to-end control it has with the iPod, and when the iPhone goes on sale in June, consumers will still have to contend with the typical cell-phone experience: the unappealing store, the confusing plan options, the two-year contract, the less-than-stellar customer service.

Oh! Did anyone remember that a new technology allows a more satisfactory resolution of a problem? And what problem was this iPhone intended to solve:

"The clever, context-based navigational system cuts out irrelevant choices and provides the intelligent and gratifying user experience we have come to expect from Apple," agrees Jakob Trollbäck, creative director of New York-based design agency Trollbäck + Co. "By eliminating intermediary input devices such as keyboard or stylus, control has become tactile again. My Blackberry Pearl has 29 keys and you need to use complicated sequences involving modifier keys to do just about anything. Getting rid of them all in one swipe, the iPhone has an interface that is digital in every sense of the word."

Trollbäck could very well be Job's prospective son-in-law, for all I know—but he does make a compelling logical point about the problem that the iPhone, as a [bundle of] technolog[ies] is supposed to solve.

One of the biggest challenges the designers faced was how to provide a suitably sized usable keyboard—something every PDA or smartphone maker has struggled with. Apple bypassed the need for a button-based keyboard by providing virtual, on-screen QWERTY keys instead—and incorporated various tricks to enhance the typing experience, such as predictive spelling and what MIT Media Lab professor John Maeda describes as the "hover-expand" behavior of the keys.

"Each key can remain small and within an orderly grid at first glance; then, by hovering your finger, the on-screen key is made bigger so that you can see it better," he explains. "It's a fairly simple idea and probably not brand new, but definitely a step forward in the awkward task of typing on a tiny virtual keyboard."

In fact, I started this blog while helping a friend who wanted to patent a new keyboard design for PDA's; and yes, one of the designs I looked at was the variable-geometry keypad.

"Scratch" is the operative word here, and concerns have already been raised about the practicality and durability of the iPhone's large, unprotected screen (and how to keep it clean). Questions also remain unanswered about the compatibility of a phone (for which battery life is paramount) and a music player (which is often used for hours at a time). Putting the two together could significantly limit a device's lifetime.

10 January 2007

Some Rueful Thoughts on Apple's iPod & iPhone

Apple has become a headache for the very artistic community that is so devoted to it. Cory Doctorow of InformationWeek writes,

The iPod is the number one music player in the world. iTunes is the No. 1 digital music store in the world. [...] But [...] no one but Apple is allowed to make players for iTunes Music Store songs, and no one but Apple can sell you proprietary file-format music that will play on the iPod. In some respects, that's not too different from other proprietary platforms, of course. No one but Microsoft makes Word. But there's a huge difference between Word and iTunes: [...] iTunes is protected by the anti-circumvention provisions in the 1998 Digital Millennium Copyright Act (DMCA), itself a law passed to comply with the 1996 UN World Intellectual Property Organization (WIPO) "Internet Treaties." The DMCA makes it a crime to circumvent "effective means of access control." That means that breaking the locks off a digital work is illegal, even if you're breaking the lock to accomplish a legal end.And as he goes on to explain, that involves reverse engineering competing products from the file formats used. Apple enjoys a unique position in that it is permanently protected from competition of this nature.Apple has already demonstrated its willingness to abuse its monopoly over iTunes players by shipping "updates" to iTunes that add new restrictions to the songs its customers have already purchased. The business model of buying music on the Internet is that one buys a "license" for certain uses, but the company that supplies the product to you can revoke parts of the license, and there's nothing you can do about it. This is just abuse.

These two things essentially make files one has stored on an iPod (including, of course, photos and recordings made with an iPhone) the effective property of Apple, not you. Porting these products to a future product is going to be difficult and perhaps even impossible or illegal.

This has a stifling effect on creativity, because when people perceive they're being taken advantage of, their response is to defy the laws entirely. Doctorow speaks of consumers voting with their wallets, but this is not about voting--it's about opting out of the putative marketplace altogether (how can there be a "market" for illegally copied audio or video?). The real analogy is not to a new electorate of passive downloading "voters," but rather, to citizens opting out of the "political system" entirely, boycotting the polls, and evading taxes en masse.

The article cited mentions the complicating factor of alternative systems of "Digital Rights Management" (DRM), or technical/legal methods of restricting downloads of audio/video to paying customers. DRMs have been a controversial subject for rival industries: movie producers and distributors, versus appliance manufacturers, versus consumers or performers. The performers are pretty much at the mercy of producers; it's difficult to come up with a viable business plan for making money performing without some gatekeeper. But producers of content, like Warner Brothers, have discovered that Apple's iTunes/iPod/iPhone series of products has become a powerful bottleneck: Apple sets the pricing structure, and producers have to put up with it. They cooperate because Apple iTunes is in many respects a more lucrative channel than record stores would be, but permission to practice discriminatory pricing would essentially allow the producers to capture a larger share of revenues from iTunes.

The main struggle to thwart Apple, by devising a rival DRM technology that is popular and conforms to the desires of producers, is described by Doctorow as a style of stuggle, and he doesn't discuss the chronology much. The desperate reliance by everyone, great and small, on monopoly control, is comically at odds with the endless references to the almighty market and the sovereign consumer.___________________________________________________UPDATE: This is not meant as a plug for Microsoft. In fact, for balance, see this article by an-ex-Windows fan who embraced Mac OS after a brush with Vista (Erika Jonietz; via Anna Ferruglia Dan).

My efforts to get Media Center working highlighted two big problems with Vista. First, it's a memory hog. The hundreds of new features jammed into it have made it a prime example of software bloat, perhaps the quintessence of programmer Niklaus Wirth's law that software gets slower faster than hardware gets faster (for more on the problems with software design that lead to bloat, see "Anything You Can Do, I Can Do Meta"). Although my computer meets the minimum requirements of a "Vista Premium Ready PC," with one gigabyte of RAM, I could run only a few ­simple programs, such as a Web browser and word processor, without running out of memory. I couldn't even watch a movie: Windows Media Player could read the contents of the DVD, but there wasn't enough memory to actually play it. In short, you need a hell of a computer just to run this OS.

08 January 2007

Constant Relative Risk Aversion

There is a game of chance called "St Petersburg," which is the simplest thing possible. Take a fair coin and flip it. You have bet 2 rubles on the outcome. If the coin comes up heads then you win 2 rubles, but if it comes up tails you play again, this time for 4 rubles. Each time, the stake is doubled, so n plays yields a prize of 2n rubles. Each flip of the coin is called a "trial" and the string of trials with their outcomes that concludes the game, is called a "consequence."

The probability of a consequence of n flips ('P(n)') is 1 divided by 2n, and the "expected payoff" of each consequence is the prize times its probability. The ‘expected value’ of the game is the sum of the expected payoffs of all the consequences. Since the expected payoff of each possible consequence is 1 ruble, and there are an infinite number of them, this sum is an infinite number of rubles. This became known as the St. Petersburg Paradox.

Bernoulli, the Swiss philosopher and mathematician, suggested the problem lay in rewarding people with money rather than utility. At the back of the paradox is the assumption that (2n)(2-n) = 1 for all values of n; as n becomes (or could become) infinitely large, the sum of probable outcomes reaches ∞. In fact, that's not true for utility, and Bernoulli proposed that the expected utility--as opposed to expected payout in rubles--was necessarily finite.1

Economists were increasingly interested in risk2 because it applies to virtually all decisions, particularly those related to savings. Suppose you have a temporary employee working at a longterm assignment. The temp could be dismissed at any second; temps are almost never given any notice, and an abrupt dismissal typically causes the temp a lot of hardship. Because of this, the temp faces a risk if she accrues any debt; savings are vital to surviving periods between assignments. Yet there is also potential benefit in taking night school courses--say, in accounting. She must therefore weigh the risk (weighted for consequences) of dismissal, against the probability of getting a permanent job with benefits (weighted for the benefits of doing so).

Putting this another way, let U be the utility experienced by the temp. U is a function of consumption C, which of course varies over time; U = U(C(t)). The temp may prefer to take risks in order to enhance her estimated future consumption: an increase in income caused by risky investment of scarce money in tuition. For small values of θ, marginal utility diminishes more slowly--i.e., U˝(C) is smaller--than for larger values. That's the crucial significance of θ.

In the equation above, a high value of θ signifies that the consumer is quickly sated by increasing consumption. Hence, both high values of consumption all at one time, and a high payoff from a high risk bear less gratification, than would be the case if θ were low. Hence, another term for "constant relative risk aversion" is "constant intertemporal elasticity of substitution" (CIES). On average, the tendency to accept risk (in exchange for a payoff) and the tendency to accept a major belt-tightening (in exchange for a future payout) are comparable.

In the graph below, the horizontal axis C(z) refers to a random outcome; the probability that z1 happens is p, and the probability that z2 happens is (1-p). In other words, either z1 or z2 can happen. So the expected outcome E(z) is pz1 + (1-p)z2. Now, please notice someone has drawn a chord between points A and B. Notice that the expected utility E(U) is substantially lower than the utility of the expected outcome u[E(z)]; or just notice D and E. The position of E on the chord is dependent on the ratio of p:(p-1).

The behavioral inference drawn from this chart is that the utility of expected income U[E(z)] is greater than the expected utiliy E(U), i.e.,

U[pz1 + (1-p)z2] > Upz1 + U(1-p)z2

This is just a complicated way of saying that risk aversion inflicts a severe hit on the utility of bundle of benefits.

The function above was developed by Milton Friedman and Leonard Savage in 1948. Friedman & Savage also speculated on other shapes of the risk-utility function, but the curve above has a certain usefulness for the economics profession. You see, if a person has a curve very much unlike the one shown above, then one can be presented with a series of risks, each of which one finds acceptable, that lead one into any position; the other party--say, the casino management--can always make a profit, and essentially "pump" money out of players. While some people undoubtedly are like that, the population in the aggregate cannot be, or the economy would grind to a halt forever.

If we are looking at the function as a CIES graph, then the horizontal access merely represents increasing values of consumption. If, however, we are looking at the function as a CRRA graph, then it makes sense to regard the horizontal axis as a series of equally likely payouts. A segment between zi and zj with a length of 1% of the entire horizontal axis, would have a 1% possibility of happening.__________________________________________________NOTES

1 For those of you unfamiliar with calculus: some algebraic functions, like f(x) = x-2 can be graphed from 0 to infinity, and the total area under their curve is finite. This seems impossible, but it's true.

2 Risk and uncertainty are (usually) regarded as distinct topics in economics. Risk is quantifiable; uncertainty is not. Or, in the words of Frank L. Knight,

The essential fact is that "risk" means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomenon depending on which of the two is really present and operating. There are other ambiguities in the term "risk" as well, which will be pointed out; but this is the most important. It will appear that a measurable uncertainty, or "risk" proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term "uncertainty" to cases of the non-quantitative type. It is this "true" uncertainty, and not risk, as has been argued, which forms the basis of a valid theory of profit and accounts for the divergence between actual and theoretical competition.[Risk, Uncertainty, and Profit, 1921]

07 January 2007

The Dynamics of Industrial Choice (2)

Industries make decisions about the implementation of technologies according to expected returns of that implementation: that's the standard position of orthodox economic theory. The reality is more complex, but it's interesting to see how even the very simple, reductionist mathematical models used to simulate the behavior of a simple economy lead to complex systems.

Since the basic rules of economic explanation are relatively simple, efforts have been made many times to reduce these to mathematical formulae that can describe the workings of a system. One idea has been to use these to re-create the laws of motion that prevail in an economy, so that the rules can be refined based on their predictive powers. The best-known attempt to do this has been the Ramsey-Cass-Koopmans Model, which simplifies the job by treating the economy as if it consisted of a single, average household.

The RAMSEY-CASS-KOOPMANS MODELAttributes of the household are inherent in the economic system. Households have endowments of labor (l) and capital (k). Labor is paid at wage w and capital commands an interest rate r. Hence, the income (y) of the household will be

y = wl + rk.

However, while the endowment k accrues interest without laobr, it also depreciates at rate δ; presumably r > δ, or else there would be little point in holding k. Likewise, L is accumulated from y - c; in the RCK model, all income that is not immediately consumed is saved, and therefore invested in the form of more k.

In economics, K always represents the total stock of capital; k (small) represents the supply available to our sample household at any moment in time. The symbol ρ stands for the discounting of future consumption; economists assume that consumers value future consumption less than present consumption.1 Stocks of capital depreciate at a rate of δ, so they must be replaced at a rate of δk. There must also be a rate of return on capital, which is r; it's common to assume that r = ρ + δ, since (by definition) if r > ρ + δ, people would be irrationally postponing consumption, and if r < ρ + δ, people are irrationally improvident. Each household is intuitively driven to maximize the equation below.

This is not as far fetched as it might seem. That's because the "average" path taken by millions of households groping towards an optimal allocation may well fit this description. Groping comes in the form of endless brushes with frustration and lost opportunities. Errors or eccentric decisions made by this or that individual may be expected to average out over very large numbers and over great lengths of time.

The household stock of capital increases at rate , which will be

= (r – δ)k + wl - c

There is a consumption function U(C) is assumed to take the form below:

This is the function for constant relative risk aversion (CRRA); it is also known as the continuous intertemporal elasticity of substitution (CIES) function. Since we do not impose a time horizon, there's a risk of what is called a "corner solution," which is where the maximum point of a function lies at one limit or the other of its domain. The danger here is that the solution would be "c = 0" for all t < ∞, since ∞ is the biggest number we have. At the end of time, k∞ would be extremely large, but the who affair would be utterly pointless since our whole effort to simulate the economy with an average household would lead to that household acting in accordance with totally arbitrary equations. Such a scenario is unreasonable; people have to consume something even when their incomes are so low they can save scarcely anything, so we have limits to the value of infinitely postponed consumption.

The economy also incorporates an average firm, which transforms l and k into y. Beyond this, however, the firm does not appear; it does not have an objective function to maximize, for instance; it is not in conflict with other firms or the representative household. The RCK is an extremely adaptive model, however, and a very large number of variations on it exist. Here, we'll be sticking to the plain vanilla version.

Click for larger image

This chart shows the two phase diagrams in the RCK model. On the left, the blue line represents constant, stable rates of consumption; c-dot represents the 1st derivative of c(t) with respect to time. Let's say that k* represents the point on the horizontal axis where c-dot = 0 (where the blue line touches the bottom edge). Then levels of capital endowment k* leads to a decrease in consumption.

(Please note that c-dot is instantaneous. I point this out because, if one occupies a position {k0, c0} , then one will presently move to another point on the phase diagram.)

On the right, the red curve indicates all the positions where k-dot is 0; if one occupies positions along that line, one's net growth in capital endowments is zero. For values of c above the red line, one's rate of capital accumulation is negative (one is spending out of one's substance!). For levels of c below the red line, one's rate of capital accumulation is positive, because one is consuming so little.

Click for larger image

Here, the two maximization functions are combined. Where the red and blue lines intersect, there is steady state consumption and capital endowment. At points along the violet line passing through the intersection, points are not in equilibrium, but are "gravitating" towards it.

The economy (in the avatar of its representative households) is has a peculiar version of the knife-edged equilibrium. The saddle equilibrium might appear to suggest that the economy, if perturbed from perfect order, would plunge into wreckage, like a locomotive on a tightrope. Over the short-run, as during recessions, this would appear to be the case; and over very long periods of history, flourishing economies do eventually enter periods of decline. However, the RCK model pertains to medium-run trends; the model is not, nor ever could be, rich enough to capture the multifarious forces of the short-run, and over the long-run things such as civil wars, obsolete institutions, demographic changes, and so forth are simply out of the model.

The other important distinction is that when an economy is not on the violet line in the graph above, the optimization preferences of its population will tend to push it toward convergence with the steady-state, balanced growth economy. In fact, the model incorporates projections of how this occurs.

In the graph above, the economy responds with a reduction in consumption, and concomitant increase in saving. Richer models incorporate a "floor" of consumption that causes it to start low, and rise to overshoot the higher balanced growth path (BGP) savings rate.

Here, the economy's stock of capital is converging. Notice that the high saving rate is accompanied by a steep slope for k; high values of s amount to exactly the same thing as high values of . Likewise, in our simplified model, the efforts of households to maximize consumption over infinite time horizons leads to rapid accumulation.

This is the big picture: the balanced growth curve, in which the endogenous factors are growing at a constant pace (dotted line) while the equilibrium growth path gradually catches up.

1 Time discounting: it is usually assumed that humans generally prefer consumption in the future to consumption in the present. As always, special exceptions may apply but remember, economists tend to be interested in average or median behavior. Even if exceptions are very common, therefore, people with unusual time-of-consumption preferences can demand the same discount as everyone else.

Typically, for purposes of government accounting the Office of Management and Budget (OMB) uses a rate of 7%, and tests for rates of 5%-9%.____________________________________________Resources and Additional Reading:

The Ramsey-Cass-Koopmans model is explained formally here (and here), for those of you interested in a backup source. The first link is to the site of Prof. Thomas M. Steger in Zurich; in my opinion, his explanation is not only the best I've seen online (and the most reliable), it's also better than the one in David Romer's textbook Advanced Macroeconomics (1st edition), which is the one I was initially using. Juan Ruiz, a Spanish economist, posted lecture notes on the RCK model that are also very easy to follow.

06 January 2007

The Dynamics of Industrial Choice (1)

Often, economic models seek to explain business decisions based on a snapshot of conditions. Hence, we have the case of the indifference curve and the production curve. Both are closely analogous:

INDIFFERENCE CURVE

In economics, one speaks of "utility" as a state that cannot be measured, but can be compared; so, for example, in the chart below, the blue line (U1) represents a lower lever level of utility than the red (U2). It is not valid to say U2 represents 1.5x as much utility, but we can include a very large number of intermediate levels of utility between the two points.

Utility is always described as a function of two sources, such as "wages" and "leisure" (from the POV of the worker). Of course, if you increase wages without reducing leisure, or increase both, then the person obviously has a higher level of utility. But what about when one must trade one for the other?

In the graph above, the economic actor's utility is a function of A and B. The red line is the result of a sudden decline in the price of B. When that happened, the "budget line"—the straight dashed lines slicing diagonally across the graph—moved outward, to the right. That diagonal line intersects the A-axis at the point where the consumer spends 100% of her income on A, and the B-axis where she spends 100% of her income on B. So when the price of B fell, the budget line moved outward to intersect with a new, higher, level of utility.

When the consumer had the lower (blue) budget line, she consumed A1 and B1. When the price of B fell, her consumption of both increased, to A2 and B2. But economists make a distinction between (A2, B2) and (A1.5, B1.5). While some of the change in consumption (ΔA,ΔB) can be explained by the increased income—i.e., the new, "purple" flashpoint on the red curve above—some of (ΔA,ΔB) is the result of substitution. So, for example, the increased real income caused by a decline in the price of B actually caused the consumption of A to decline in absolute terms. An income effect will always cause both to increase, but a substitution effect will always cause consumption of one to fall relative to the other.

OUTPUT CURVE

This is closely analogous to the indifference curve, and so I used a similar graphic with different labels (the original graphic is here).

Here, the tradeoff is between labor (L) and capital (K), or any other combination of inputs. While I've shown only two inputs in the diagram, it's possible to set up optimization equations involving as many inputs as you like... such as different capital structures (bonds versus bank loans versus equity), energy inputs, and so forth. One element that is new to the production curve here is the idea of technology: the possibility that output (X) can increase without an increase in L or K. In fact, economists simply treat technology as another input (A), and have long debated the role it plays (here's a formal treatment).

The concept of the indifference curve in economics dates back to the 1870's; some of the first economists to use it were the "Marginalists," such as William Stanley Jevons (1871) and Leon Walras (1874). A formal explanation of these concepts may be found here.

PARETO OPTIMIZATION

Pareto Optimization is illustrated by the Edgeworth Box shown below. It's really just a pair of indifference curves. One thing to remember is that, while the vertical axis shows rising wages, the direction of the horizontal axis is reversed. That's because the "zero" axis for the utility of the owner of capital is in the extreme upper right-hand corner of the graph.

According to this chart, the rising rate of wages is one contributor to the utility of the worker; another contribution is lower interest rates (or capital rental rates). The latter effectively increases the purchasing power of the worker.

(Incidentally, this is not a radical or leftist conception of labor-capital relations. It's from the work of Francis Ysidro Edgeworth and Vilfredo Pareto, two of the most conservative, orthodox economists who ever lived. Anyone who is seriously disturbed by my dichotomy can relax. Everyone agrees that there's a missing dimension here, which is that of time. A reasonably high value of r leads to an increase in the accumulation of capital, allowing for greater total output.)

The object of this diagram was to illustrate how the market, under optimal conditions, resolves the controversy of the correct distribution of the total economic output between labor and capital. This same dichotomy of interest also exists between producers and suppliers, or taxpayers and the state.

However, the chart also illustrates something else: one can see here the idea of demand reaching a convergence with available output. The optimal solution is one where the rate of indifference is the same as the comparative cost, which is (in turn) determined by the output function.